Math’s Block-Stacking Problem Has a Preposterous Solution

This Block-Stacking Math Problem Has a Preposterous Solution You Need to See to Believe

By Jack Murtagh edited by Jeanna Bryner

Here’s a mind-blowing experiment that you can try at home: Gather some children’s blocks and place them on a table. Take one block and slowly push it over the table’s edge, inch by inch, until it’s on the brink of falling. If you possess patience and a steady hand, you should be able to balance it so that exactly half of it hangs off the edge. Nudge it any farther, and gravity wins. Now take two blocks and start over. Stacking one on top of the other, how far can you get the end of the top block to poke over the table’s edge?

Keep going. Stacking as many blocks as you can, what is the farthest overhang you can achieve before the whole structure topples? Is it possible for the tower to extend a full block length beyond the lip of the table? Does physics permit two block lengths? The stunning answer is that the stacked bridge can stretch forever. In principle, a freestanding stack of blocks can span the Grand Canyon, no glue required.

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Don’t click “checkout” on an infinite pack of Jenga blocks just yet. Real-world practicalities such as irregular block shapes, air currents and the crushing weight of an endless edifice may hamper your engineering aspirations. Still, understanding why the overhang has no limit in an ideal mathematical world is enlightening. The explanation hinges on math’s harmonic series and the physics concept of center of mass, two seemingly simple ideas with outsize power. [For more fun, check out: How Tall Can You Build a Tower without It Toppling?]

Your intuition might tell you that a single block can hang half of its mass beyond the table’s edge before tipping. But why is that so? Every object has a center of mass—a single point at which we can imagine the entire object’s weight to be concentrated when we’re thinking about balance. As long as the center of mass sits above the table, the object stays put. The moment that center of mass passes over an edge, however, gravity will…